JAX implementations of various elliptic integrals.
Forward and reverse mode autodiff compatible
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$el1(x, k_c)$ : incomplete elliptic integral of the first kind -
$el2(x, k_c, a, b)$ : incomplete elliptic integral of the second kind -
$el3(x, k_c, p)$ : incomplete elliptic integral of the third kind -
$cel(k_c, p, a, b)$ : Generalized complete elliptic integral
Forward mode autodiff only
$R_\mathrm{F}(x, y, z)$ $R_\mathrm{C}(x, y)$ $R_\mathrm{J}(x, y, z, p)$ $R_\mathrm{D}(x, y, z)$
Forward mode autodiff only
-
$K(k)$ : complete elliptic integral of the first kind -
$E(k)$ : complete elliptic integral of the second kind -
$\Pi(n, k)$ : complete elliptic integral of the third kind -
$F(\phi, k)$ : incomplete elliptic integral of the first kind -
$E(\phi, k)$ : incomplete elliptic integral of the second kind -
$\Pi(\phi, k, n)$ : incomplete elliptic integral of the third kind
Note: The Legendre forms are computed directly from the Carlson integrals using the relations found in [3].
[1] Bulirsch, 1969b
[2] Carlson, 1994